Mathematics > Algebraic Geometry

Abstract: The first part of this work constructs real positive-genus Gromov-Witten
invariants of real-orientable symplectic manifolds of odd "complex" dimensions;
the second part studies the orientations on the moduli spaces of real maps used
in constructing these invariants. The present paper applies the results of the
latter to obtain quantitative and qualitative conclusions about the invariants
defined in the former. After describing large collections of real-orientable
symplectic manifolds, we show that the real genus 1 Gromov-Witten invariants of
sufficiently positive almost Kahler threefolds are signed counts of real genus
1 curves only and thus provide direct lower bounds for the counts of these
curves in such targets. We specify real orientations on the real-orientable
complete intersections in projective spaces; the real Gromov-Witten invariants
they determine are in a sense canonically determined by the complete
intersection itself, (at least) in most cases. We also obtain equivariant
localization data that computes the real invariants of projective spaces and
determines the contributions from many torus fixed loci for other complete
intersections. Our results confirm Walcher's predictions for the vanishing of
these invariants in certain cases and for the localization data in other cases.